Theorem exmidontri2or 7592

Description: Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)

Assertion

Ref	Expression
exmidontri2or	|- (EXMID <-> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )

Distinct variable group:   x, y

Proof of Theorem exmidontri2or

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exmidontriim 7571	. . 3 |- (EXMID -> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )
2		onelss 4527	. . . . . . . 8 |- ( y e. On -> ( x e. y -> x C_ y ) )
3	2	adantl 277	. . . . . . 7 |- ( ( x e. On /\ y e. On ) -> ( x e. y -> x C_ y ) )
4		orc 724	. . . . . . 7 |- ( x C_ y -> ( x C_ y \/ y C_ x ) )
5	3, 4	syl6 33	. . . . . 6 |- ( ( x e. On /\ y e. On ) -> ( x e. y -> ( x C_ y \/ y C_ x ) ) )
6		eqimss 3302	. . . . . . . 8 |- ( x = y -> x C_ y )
7	6, 4	syl 14	. . . . . . 7 |- ( x = y -> ( x C_ y \/ y C_ x ) )
8	7	a1i 9	. . . . . 6 |- ( ( x e. On /\ y e. On ) -> ( x = y -> ( x C_ y \/ y C_ x ) ) )
9		onelss 4527	. . . . . . . 8 |- ( x e. On -> ( y e. x -> y C_ x ) )
10	9	adantr 276	. . . . . . 7 |- ( ( x e. On /\ y e. On ) -> ( y e. x -> y C_ x ) )
11		olc 723	. . . . . . 7 |- ( y C_ x -> ( x C_ y \/ y C_ x ) )
12	10, 11	syl6 33	. . . . . 6 |- ( ( x e. On /\ y e. On ) -> ( y e. x -> ( x C_ y \/ y C_ x ) ) )
13	5, 8, 12	3jaod 1345	. . . . 5 |- ( ( x e. On /\ y e. On ) -> ( ( x e. y \/ x = y \/ y e. x ) -> ( x C_ y \/ y C_ x ) ) )
14	13	ralimdva 2617	. . . 4 |- ( x e. On -> ( A. y e. On ( x e. y \/ x = y \/ y e. x ) -> A. y e. On ( x C_ y \/ y C_ x ) ) )
15	14	ralimia 2611	. . 3 |- ( A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) -> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )
16	1, 15	syl 14	. 2 |- (EXMID -> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )
17		ontri2orexmidim 4714	. . . 4 |- ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) -> DECID z = { (/) } )
18	17	adantr 276	. . 3 |- ( ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) /\ z C_ { (/) } ) -> DECID z = { (/) } )
19	18	exmid1dc 4332	. 2 |- ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) -> EXMID )
20	16, 19	impbii 126	1 |- (EXMID <-> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   \/ w3o 1008   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   (/)c0 3520   {csn 3705  EXMIDwem 4326   Oncon0 4503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-tr 4225  df-exmid 4327  df-iord 4506  df-on 4508  df-suc 4511

This theorem is referenced by:  onntri52  7593